I know you said it wasn't essential, but what is you're reasoning on swapping some of those? Rows and dips for instance: I figured after the bench press, I would be fresher for the dips if they were separated by rows since rows are more back dominant while bench and dips are both chest and triceps dominant. Same thing with good mornings, which are lower back dominant like deadlifts, while chins are not.

As it turns out, getting 100 body weight chins total for the week as recommended by the 5/3/1 book is going to be more difficult than I thought. Obviously, I knew I couldn't do it in 5x10 fashion, hence why I said (or whatever it takes to reach 50 reps). But today was bench day. I switched rows with dips and did the dips first. When come time to do rows, all the barbells are being used. So I chose chinups instead. Maxing out 5 sets, I was only able to get to 27 total reps. This is with 3 minute rests. The first couple sets weren't too bad, but after that, I was only getting about 4 reps before failure. I definitely don't have time to get to 50 reps. I know with assistance, I'm not supposed to think about it too much, but I've concluded that upper back strength has always been my weak point and I want to get better. Thought?

I would just do the five sets and try to get as many as possible, and then try to improve that number from week to week.
Or change to weighted chins and just go for three sets of five.
I do weighted chins and bw dips on press day, and bw pullups and weighted dips on bench day. So that's also an option.